Arithmetic Progressions

An arithmetic progression (AP), also known as an arithmetic sequence, is a sequence of numbers such that the difference between any two consecutive terms is constant. This constant difference is called the common difference, often denoted by the letter ’d'.

The following diagrams give an arithmetic progression and the formulas to find the n^th term and the sum of the first n terms. Scroll down the page for more examples and solutions.

An arithmetic progression can be written in the form:
a, a + d, a + 2d, a + 3d, … , a + (n-1)d, …
Where:
‘a’ is the first term of the sequence.
’d’ is the common difference.
’n’ is the position of a term in the sequence (e.g., for the 3rd term, n = 3).
‘a + (n-1)d’ is the formula for the n-th term of the sequence (often denoted as a_n or T_n).

Key Formulas Related to Arithmetic Progressions:

1. n-th term (a_n or T_n)
   a_n = a + (n − 1)d
   
2. Common Difference (d):
   d = a_n - a_n-1
   This means you can find the common difference by subtracting any term from its succeeding term.
   
3. Sum of the first n terms (S_n):
   There are two common formulas for the sum:
   

• When you know the first term (a), the common difference (d), and the number of terms (n):
  \(S_n = \frac{n}{2}[2a+(n-1)d]\)
  
• When you know the first term (a), the last term (l or a_n), and the number of terms (n):
  \(S_n = \frac{n}{2}[a+l]\)
  

Arithmetic Progressions : Introduction
In this video tutorial you are introduced to what an arithmetic progression is and how to find the nth term and the sum of the first n terms.

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Arithmetic Progressions : Proof of the Sum of n terms
This video will show you how to prove the formulae for the sum S(n) of the first n terms of an arithmetic progression which is often asked in exams.

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Arithmetic Series : C1 Edexcel January 2013 Q7c
7. Sian played an adventure game. She scored points for each dragon that she captured. The number of points that Sian scored for capturing each successive dragon formed an arithmetic sequence. Sian captured n dragons and the total number of points that she scored for capturing all n dragons was 8500. Given that Sian scored 300 points for capturing her first dragon and then 700 points for capturing her nth dragon, (c) find the value of n

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How to prove the sum of n terms of an arithmetic series?

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